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uzerr
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Difference between 2 types of probability questions Updated on: 13 Dec 2025, 05:30
Originally posted by uzerr on 13 Dec 2025, 05:24.
Last edited by uzerr on 13 Dec 2025, 05:30, edited 1 time in total.
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1st question- 9 people in room. 2 pairs of siblings within that group. If two individuals are selected from the room, what's the probability they're NOT siblings?
p= 4/9 * 7/8 + 5/9 * 8/8 = 17/18

2nd question- 7 people in a room, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings?
p= 2 * 3/7 * 4/6 + 2 * 2/7 * 2/6 = 16/21

Why are 2's multiplied in the 2nd problem but not the 1st problem?
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Difference between 2 types of probability questions 13 Dec 2025, 05:30
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The difference is not a conceptual one about siblings, but about the method of counting :
the second solution conditions on ordered selections (requiring a factor of 2 for symmetry), while the first problem is solved using unordered pairs, where no such factor is needed
uzerr
1st question- 9 people in room. 2 pairs of siblings within that group. If two individuals are selected from the room, what's the probability they're NOT siblings?
p= 4/9 * 7/8 + 5/9 * 8/8 = 17/18

2nd question- 7 people in a room, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings?
p= 2 * 3/7 * 4/6 + 2 * 2/7 * 2/6 = 16/21

Why is 2 multiplied in the 2nd problem but not the 1st problem?
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Difference between 2 types of probability questions 13 Dec 2025, 05:39
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uzerr
1st question- 9 people in room. 2 pairs of siblings within that group. If two individuals are selected from the room, what's the probability they're NOT siblings?
p= 4/9 * 7/8 + 5/9 * 8/8 = 17/18

2nd question- 7 people in a room, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings?
p= 2 * 3/7 * 4/6 + 2 * 2/7 * 2/6 = 16/21

Why are 2's multiplied in the 2nd problem but not the 1st problem?
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The point is that the approaches used for the first and second questions are different. We can solve the second question the same way as the first, and then we won’t need to multiply by 2: 4/7 * 5/6 + 3/7 * 4/6 = 16/21.

The reason in that approach for the second question * 2 is used is explained here: https://gmatclub.com/forum/in-a-room-fi ... l#p1096494

1st question is discussed in detail here: https://gmatclub.com/forum/there-are-9- ... 58609.html
2nd question is discussed in detail here: https://gmatclub.com/forum/in-a-room-fi ... 87550.html

Hope it helps.
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Difference between 2 types of probability questions 13 Dec 2025, 06:08
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Thank you for the explanation. I have read those threads but was still unsure about the exact methods.

Could the 1st problem be done as the 2nd like this? : 2 * 2/9*2/8 + 2 *5/9*6/8= 17/18
Bunuel


The point is that the approaches used for the first and second questions are different. We can solve the second question the same way as the first, and then we won’t need to multiply by 2: 4/7 * 5/6 + 3/7 * 4/6 = 16/21.

The reason in that approach for the second question * 2 is used is explained here: https://gmatclub.com/forum/in-a-room-fi ... l#p1096494

1st question is discussed in detail here: https://gmatclub.com/forum/there-are-9- ... 58609.html
2nd question is discussed in detail here: https://gmatclub.com/forum/in-a-room-fi ... 87550.html

Hope it helps.
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Difference between 2 types of probability questions 13 Dec 2025, 06:09
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uzerr
Thank you for the explanation. I have read those threads but was still unsure about the exact methods.

Could the 1st problem be done as the 2nd like this? : 2 * 2/9*2/8 + 2 *5/9*6/8= 17/18

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Yes, but that's not the best approach. Check and study other ones.
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Difference between 2 types of probability questions 13 Dec 2025, 07:33
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Thanks for the explanation. Is there a way of determining when to use ordered vs unordered selections? When I try to do Q1 like Q2 (3 groups- 2 pairs siblings, 1 group of 5 with no siblings), I get 2 * 2/9*2/8 + 2 *5/9*8/8= 88/72 which is completely wrong

Knowing the answer is 17/18, I assume it would have to be 2 * 2/9*2/8 + 2 *5/9*6/8= 68/72. But I'm not sure what 6/8 would represent.

I have read the threads on these problems, which give solutions which I can understand. But I still can't pinpoint when to use unordered vs ordered selections, and why using ordered pairs on Q1 didn't work.
annamariaki
The difference is not a conceptual one about siblings, but about the method of counting :
the second solution conditions on ordered selections (requiring a factor of 2 for symmetry), while the first problem is solved using unordered pairs, where no such factor is needed

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Difference between 2 types of probability questions 11 Jan 2026, 13:16
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I don’t think these are good examples for ordered vs. unordered selections. Ordered selections are mainly used when you actually care about the order in which items are selected in a sequence. Also, honestly, I’d recommend learning the counting method instead of relying on this probability-based approach, because that can get confusing in more complex scenarios. Like if I have to attempt these, I will do something like -

1. 9 people & 2 pair of siblings

Total ways to select 2 people = 9C2 = 9x8/2 = 36
How many ways to select 2 siblings = 2

P(not siblings) = 1- P(siblings) = 1 - 2/36 = 17/18

2. 7 people & 2 pair of siblings & 1 pair of triplets

Total ways to select 2 people = 7C2 = 7x6/2 = 21
Total ways to select 2 siblings = 2 + 3C2 = 5

P(not siblings) = 1- P(siblings) = 1 - 5/21 = 16/21

Benefit of doing it this way is that you can always count the combinations which you are including or excluding, and hence you won't end up in the ordered or unordered trap which would be difficult for you to verify in the end when you are solving it under the clock.
uzerr
Thanks for the explanation. Is there a way of determining when to use ordered vs unordered selections? When I try to do Q1 like Q2 (3 groups- 2 pairs siblings, 1 group of 5 with no siblings), I get 2 * 2/9*2/8 + 2 *5/9*8/8= 88/72 which is completely wrong

Knowing the answer is 17/18, I assume it would have to be 2 * 2/9*2/8 + 2 *5/9*6/8= 68/72. But I'm not sure what 6/8 would represent.

I have read the threads on these problems, which give solutions which I can understand. But I still can't pinpoint when to use unordered vs ordered selections, and why using ordered pairs on Q1 didn't work.

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